To solve the given systems of equations, we need to determine the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy each system. Let's solve each system one by one:
### System 1:
[tex]\[
\begin{cases}
x + 8y = -25 \\
-x + 3y = -8
\end{cases}
\][/tex]
To solve this system, we can add the two equations to eliminate [tex]\(x\)[/tex]:
[tex]\[
(x + 8y) + (-x + 3y) = -25 + (-8)
\][/tex]
[tex]\[
11y = -33
\][/tex]
[tex]\[
y = -3
\][/tex]
Now substitute [tex]\(y = -3\)[/tex] back into the first equation:
[tex]\[
x + 8(-3) = -25
\][/tex]
[tex]\[
x - 24 = -25
\][/tex]
[tex]\[
x = -1
\][/tex]
Thus, the solution for the first system is:
[tex]\[
(x, y) = (-1, -3)
\][/tex]
### System 2:
[tex]\[
\begin{cases}
7x + 2y = -3 \\
21x + 6y = -3
\end{cases}
\][/tex]
Notice that the second equation is just a multiple of the first equation. Specifically, the second equation can be rewritten as:
[tex]\[
3(7x + 2y) = -3 \implies 21x + 6y = -3
\][/tex]
This indicates that the two equations are essentially the same, which means they represent the same line. Consequently, there are infinitely many solutions for this system that lie on this line. We denote this as:
[tex]\[
\text{Solution} = \infty \text{ many solutions, } oo
\][/tex]
### System 3:
[tex]\[
\begin{cases}
5x + 5y = 75 \\
-5x + 4y = -3
\end{cases}
\][/tex]
First, simplify the first equation by dividing by 5:
[tex]\[
x + y = 15
\][/tex]
Now solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = 15 - y
\][/tex]
Substitute [tex]\(x = 15 - y\)[/tex] into the second equation:
[tex]\[
-5(15 - y) + 4y = -3
\][/tex]
[tex]\[
-75 + 5y + 4y = -3
\][/tex]
[tex]\[
9y = 72
\][/tex]
[tex]\[
y = 8
\][/tex]
Substitute [tex]\( y = 8 \)[/tex] back into the equation [tex]\(x + y = 15\)[/tex]:
[tex]\[
x + 8 = 15
\][/tex]
[tex]\[
x = 7
\][/tex]
Thus, the solution for the third system is:
[tex]\[
(x, y) = (7, 8)
\][/tex]
### System 4:
[tex]\[
\begin{cases}
2x - 2y = 5 \\
-6x + 6y = -15
\end{cases}
\][/tex]
Simplify each equation by dividing the first by 2 and the second by -6:
[tex]\[
x - y = \frac{5}{2}
\][/tex]
[tex]\[
x - y = \frac{5}{2}
\][/tex]
This shows that both equations are the same line. Thus, there are infinitely many solutions along the line:
[tex]\[
x = y + \frac{5}{2}
\][/tex]
Putting it all together, the solutions for each system are:
1. [tex]\( (-1, -3) \)[/tex]
2. [tex]\(oo\)[/tex]
3. [tex]\( (7, 8) \)[/tex]
4. [tex]\( x = y + \frac{5}{2} \)[/tex]
